How do I calculate the gradient of the hinge loss function?
Hinge loss is difficult to work with when the derivative is needed because the derivative will be a piece-wise function. max has one non-differentiable point in its solution, and thus the derivative has the same. This was a very prominent issue with non-separable cases of SVM (and a good reason to use ridge regression).

Here's a slide (Original source from Zhuowen Tu, apologies for the title typo):


Where hinge loss is defined as max(0, 1-v) and v is the decision boundary of the SVM classifier. More can be found on the Hinge Loss Wikipedia.

As for your equation: you can easily pick out the v of the equation, however without more context of those functions it's hard to say how to derive. Unfortunately I don't have access to the paper and cannot guide you any further...